1. 1
   A school debating society has 7 members in Year 12 and 5 members in Year 13. A team of 4 members is to be chosen from the society to enter a competition. The order in which the team members are chosen does not matter.

   (a)

   Find the number of different teams that can be chosen.

   [1 mark]

   • A$11\,880$
   • B$495$
   • C$20\,736$
   • D$210$

   (b)

   Find the number of teams that contain exactly two Year 12 members and exactly two Year 13 members.

   [1 mark]

   • A$31$
   • B$840$
   • C$66$
   • D$210$

   (c)

   Find the number of teams that contain at least one Year 13 member.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f(x) = (1 - 2x)^{-3}$ is expanded as a series in ascending powers of $x$.

   (a)

   Find the coefficient of $x$ in the expansion.

   [1 mark]

   • A$-6$
   • B$-2$
   • C$6$
   • D$-3$

   (b)

   Find the coefficient of $x^{2}$ in the expansion.

   [1 mark]

   • A$24$
   • B$12$
   • C$48$
   • D$-24$

   (c)

   State the set of values of $x$ for which the expansion is valid.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A security PIN consists of four different digits chosen from the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 (zero is not used and no digit may be repeated). The order of the digits matters, so 2791 and 1279 are different PINs.

   (a)

   Find the number of possible PINs that, when read as a four-digit number, are odd.

   [3 marks]

   (b)

   Find the number of possible PINs whose digits are in strictly increasing order from left to right. Hence find the probability that a PIN chosen at random from all possible PINs has its digits in strictly increasing order.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $g(x) = (8 + 3x)^{\frac{1}{3}}$.

   (a)

   Show that $g(x) = 2\left(1 + \dfrac{3x}{8}\right)^{\frac{1}{3}}$. Hence find the first three terms of the expansion of $g(x)$ in ascending powers of $x$, and state the set of values of $x$ for which the expansion is valid.

   [6 marks]

   (b)

   The expansion of $(1 + kx)\,g(x)$ in ascending powers of $x$ has no term in $x^{2}$.
   (i) Find the value of $k$.
   (ii) Find the coefficient of $x$ in the expansion of $(1 + kx)\,g(x)$.
   (iii) State, with a reason, the set of values of $x$ for which the expansion of $(1 + kx)\,g(x)$ is valid.

   [6 marks]

   Total for question 4: 12 marks

End of questions